Optimal. Leaf size=134 \[ \frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(d+e x) (5 x (2203 d+8553 e)+34347 d-6413 e)}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^2}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{e (40 d-49 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{4 e^2 x}{125} \]
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Rubi [A] time = 0.238742, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1644, 1657, 634, 618, 204, 628} \[ \frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(d+e x) (5 x (2203 d+8553 e)+34347 d-6413 e)}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^2}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{e (40 d-49 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{4 e^2 x}{125} \]
Antiderivative was successfully verified.
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Rule 1644
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^3} \, dx &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{(d+e x) \left (\frac{2}{125} (3267 d+2734 e)-\frac{6}{125} (3080 d-1371 e) x+\frac{112}{25} (20 d-33 e) x^2+\frac{448 e x^3}{5}\right )}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{\frac{4}{125} \left (42375 d^2-55870 d e+6413 e^2\right )+\frac{6272}{125} (40 d-41 e) e x+\frac{25088 e^2 x^2}{25}}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{25088 e^2}{125}+\frac{4 \left (42375 d^2-55870 d e-12403 e^2+1568 (40 d-49 e) e x\right )}{125 \left (3+2 x+5 x^2\right )}\right ) \, dx}{6272}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{42375 d^2-55870 d e-12403 e^2+1568 (40 d-49 e) e x}{3+2 x+5 x^2} \, dx}{196000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{((40 d-49 e) e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{1250}+\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{980000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{(40 d-49 e) e \log \left (3+2 x+5 x^2\right )}{1250}+\frac{\left (-211875 d^2+342070 d e-14817 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{490000}\\ &=\frac{4 e^2 x}{125}-\frac{(1367+423 x) (d+e x)^2}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x) (34347 d-6413 e+5 (2203 d+8553 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{980000 \sqrt{14}}+\frac{(40 d-49 e) e \log \left (3+2 x+5 x^2\right )}{1250}\\ \end{align*}
Mathematica [A] time = 0.176357, size = 146, normalized size = 1.09 \[ \frac{70 \left (\frac{5 \left (5 d^2 \left (11015 x^3+38753 x^2+17979 x+12953\right )+2 d e \left (181765 x^3+28307 x^2+57761 x-19533\right )+e^2 \left (156800 x^5+125440 x^4+83809 x^3-138345 x^2-65427 x-76977\right )\right )}{\left (5 x^2+2 x+3\right )^2}+784 e (40 d-49 e) \log \left (5 x^2+2 x+3\right )\right )+5 \sqrt{14} \left (211875 d^2-342070 d e+14817 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{68600000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 179, normalized size = 1.3 \begin{align*}{\frac{4\,{e}^{2}x}{125}}+{\frac{1}{5\, \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ( \left ({\frac{2203\,{d}^{2}}{1568}}+{\frac{36353\,de}{3920}}-{\frac{129439\,{e}^{2}}{39200}} \right ){x}^{3}+ \left ({\frac{38753\,{d}^{2}}{7840}}+{\frac{28307\,de}{19600}}-{\frac{213609\,{e}^{2}}{39200}} \right ){x}^{2}+ \left ({\frac{17979\,{d}^{2}}{7840}}+{\frac{57761\,de}{19600}}-{\frac{4875\,{e}^{2}}{1568}} \right ) x+{\frac{12953\,{d}^{2}}{7840}}-{\frac{19533\,de}{19600}}-{\frac{76977\,{e}^{2}}{39200}} \right ) }+{\frac{4\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) de}{125}}-{\frac{49\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{2}}{1250}}+{\frac{339\,\sqrt{14}{d}^{2}}{21952}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{34207\,\sqrt{14}de}{1372000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{14817\,\sqrt{14}{e}^{2}}{13720000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54316, size = 209, normalized size = 1.56 \begin{align*} \frac{4}{125} \, e^{2} x + \frac{1}{13720000} \, \sqrt{14}{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{1250} \,{\left (40 \, d e - 49 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{{\left (55075 \, d^{2} + 363530 \, d e - 129439 \, e^{2}\right )} x^{3} +{\left (193765 \, d^{2} + 56614 \, d e - 213609 \, e^{2}\right )} x^{2} + 64765 \, d^{2} - 39066 \, d e - 76977 \, e^{2} +{\left (89895 \, d^{2} + 115522 \, d e - 121875 \, e^{2}\right )} x}{196000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25355, size = 933, normalized size = 6.96 \begin{align*} \frac{10976000 \, e^{2} x^{5} + 8780800 \, e^{2} x^{4} + 70 \,{\left (55075 \, d^{2} + 363530 \, d e + 83809 \, e^{2}\right )} x^{3} + 70 \,{\left (193765 \, d^{2} + 56614 \, d e - 138345 \, e^{2}\right )} x^{2} + \sqrt{14}{\left (25 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{4} + 20 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{3} + 34 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x^{2} + 1906875 \, d^{2} - 3078630 \, d e + 133353 \, e^{2} + 12 \,{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 4533550 \, d^{2} - 2734620 \, d e - 5388390 \, e^{2} + 70 \,{\left (89895 \, d^{2} + 115522 \, d e - 65427 \, e^{2}\right )} x + 10976 \,{\left (25 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{4} + 20 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{3} + 34 \,{\left (40 \, d e - 49 \, e^{2}\right )} x^{2} + 360 \, d e - 441 \, e^{2} + 12 \,{\left (40 \, d e - 49 \, e^{2}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{13720000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.38472, size = 304, normalized size = 2.27 \begin{align*} \frac{4 e^{2} x}{125} + \left (\frac{e \left (40 d - 49 e\right )}{1250} - \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{27440000}\right ) \log{\left (x + \frac{42375 d^{2} - 244030 d e + 218093 e^{2} + \frac{21952 e \left (40 d - 49 e\right )}{5} - \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{5}}{211875 d^{2} - 342070 d e + 14817 e^{2}} \right )} + \left (\frac{e \left (40 d - 49 e\right )}{1250} + \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{27440000}\right ) \log{\left (x + \frac{42375 d^{2} - 244030 d e + 218093 e^{2} + \frac{21952 e \left (40 d - 49 e\right )}{5} + \frac{\sqrt{14} i \left (211875 d^{2} - 342070 d e + 14817 e^{2}\right )}{5}}{211875 d^{2} - 342070 d e + 14817 e^{2}} \right )} + \frac{64765 d^{2} - 39066 d e - 76977 e^{2} + x^{3} \left (55075 d^{2} + 363530 d e - 129439 e^{2}\right ) + x^{2} \left (193765 d^{2} + 56614 d e - 213609 e^{2}\right ) + x \left (89895 d^{2} + 115522 d e - 121875 e^{2}\right )}{4900000 x^{4} + 3920000 x^{3} + 6664000 x^{2} + 2352000 x + 1764000} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14012, size = 194, normalized size = 1.45 \begin{align*} \frac{1}{13720000} \, \sqrt{14}{\left (211875 \, d^{2} - 342070 \, d e + 14817 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{4}{125} \, x e^{2} + \frac{1}{1250} \,{\left (40 \, d e - 49 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{{\left (55075 \, d^{2} + 363530 \, d e - 129439 \, e^{2}\right )} x^{3} +{\left (193765 \, d^{2} + 56614 \, d e - 213609 \, e^{2}\right )} x^{2} + 64765 \, d^{2} +{\left (89895 \, d^{2} + 115522 \, d e - 121875 \, e^{2}\right )} x - 39066 \, d e - 76977 \, e^{2}}{196000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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